Self-intersections of trajectories of Lorentz process

We study the asymptotic behaviour of the number of self-intersections of a trajectory of a periodic planar Lorentz process with strictly convex obstacles and finite horizon. We give precise estimates for its expectation and its variance. As a consequence, we establish the almost sure convergence of the self-intersections with a suitable normalization.


Introduction
The Lorentz process describes the evolution of a point particle moving at unit speed in a domain Q with elastic reflection on ∂Q. We consider here a planar Lorentz process in a Z 2 -periodic domain Q ⊆ R 2 with strictly convex obstacles U i,ℓ constructed as follows. We choose a finite number of convex open sets O 1 , ..., O I ⊂ R 2 with C 3 -smooth boundary and with non null curvature. We repeat these sets Z 2 -periodically by defining U i,ℓ = O i + ℓ for every (i, ℓ) ∈ {1, ..., I} × Z 2 . We suppose that the closures of the U i,ℓ are pairwise disjoint. Now we define the domain Q := R 2 \ I i=1 ℓ∈Z 2 U i,ℓ . We assume that the horizon is finite, which means that every line meets the boundary of Q (i.e. there is no infinite free flight). We consider a point particle moving in Q with unit speed and with respect to the Descartes reflection law at its reflection times (reflected angle=incident angle). We call configuration of a particle at some time the couple constituted by its position and its speed. The Lorentz process in the domain Q is the flow (Y t ) t on Q × S 1 such that Y t maps the configuration at time 0 to the configuration at time t. We assume that the initial distribution P is uniform on (Q ∩ [0, 1] 2 ) × S 1 . The study of the Lorentz process is strongly related to the corresponding Sinai billiard (M ,μ,T ). Recall that this billiard is the probability dynamical system describing the dynamics of the Lorentz process modulo Z 2 and at reflection times. Ergodic properties of this dynamical system have been studied namely by Sinai in [19] (for its ergodicity), Bunimovich and Sinai [2,3], Bunimovich, Chernov and Sinai [4,5] (for central limit theorems), Young [21] (for exponential rate of decorrelation). Other limit theorems for the Sinai billiard and its applications to the Lorentz process have been investigated in many papers, let us mention namely [7,14,20] for its ergodicity and [9] for some other properties.
We are interested here in the study of the following quantity, called number of selfintersections of the trajectory of the Lorentz Process: where π Q denotes the canonical projection from Q × S 1 to Q (i.e. π Q (q, v) = q). This quantity V t corresponds to the number of couples of times (r, s) before time t such that the particle was at the same position in the plane at both times r and s. We also define V n as the number of self-intersections up to the nth reflection time. The studies of V t and of V n are naturally linked.
Self-intersections of random walks have been studied by many authors (see [6] and references therein). Motivated by the study of planar random walks in random sceneries, Bolthausen [1] established an exact estimate for the expectation of the number of self-intersections of planar recurrent random walks. He also stated an upper bound for its variance. This last estimate was sufficient for his purpose but not optimal. A precise estimate for this variance has recently been stated by Deligiannidis and Utev [8].
In view of planar Lorentz process in random scenery, another notion of self-intersections of Lorentz process arises: the number of self-intersections of the Lorentz process seen on obstacles, i.e. the numberV n of couples of times (r, s) (before the n-th reflection) such that the particle hit the same obstacle at both times r and s. This quantity has been studied in [16,17]. In the present work, our approach has some common points with [16,17] but the study of V n (and thus of V t ) is much more delicate than the study ofV n (see section 2 for some explanations).
Let us define (I k , S k ) in {1, ..., I} × Z 2 as the index of the obstacle hit at the k-th reflection time ((I 0 , S 0 ) being the index of the obstacle at time 0 or at the last reflection time before 0). The asymptotic behaviour of (S n ) n plays some role here. In particular, our proofs use a decorrelation result and some precised local limit theorems for (S n ) n . As a consequence, the constants appearing in our statements are expressed in terms of the asymptotic (positive) variance matrix Σ 2 of (k −1/2 S k ) k≥1 (with respect toμ).

Theorem 1. We have
where τ is the free flight length until the next reflection time.
Let us indicate that these results are generalized in Corollaries 15 and 17 to a wider class of initial probability measures.
and J := Corollary 4. The following convergences hold almost everywhere (with respect toμ and to the Lebesgue measure on Q × S 1 respectively): The paper is organized as follows. In Section 1, we introduce the billiard systems, some notations and local limit theorems with remainder terms. In Section 2, we prove Theorem 1. In Section 3, we establish a decorrelation result in view of our proof of Theorem 3 in Section 4. In Section 5, we use Theorems 1 and 3 to prove Theorem 2 and some generalization of Theorems 1 and 2 to a class of probability measures. Finally we prove Corollary 4 in Section 6.

Lorentz process and billiard systems
We denote by ·, · the usual scalar product on R 2 and by | · | the supremum norm on R 2 .
1.1. planar billiard system. For any q ∈ ∂Q, we write n q for the unit normal vector to ∂Q at q directed into Q. We consider the set M of couples position-unit speed (q, v) corresponding to a reflected vector on ∂Q: • q − ℓ is the point of ∂O i with curvilinear absciss r for the trigonometric orientation (starting from q i ) • ϕ is the angular measure of ( n q , v).
We consider the transformation T mapping a reflected vector to the reflected vector corresponding to the next collision time. T preserves the (infinite) measure µ with density cos(ϕ) with respect to the measure drdϕ on M . This infinite measure dynamical system (M, µ, T ) is called planar billiard system. We endow M with a metric d equal to max(|r − r ′ |, |ϕ − ϕ ′ |) on any obstacle ∂U i,ℓ . We define the map τ : which corresponds to the length of the free flight of a particle starting from q with initial speed v. Due to our assumptions, we have min τ > 0 and max τ < ∞.
We define R 0 as the set of (q, v) ∈ M with v tangent to ∂Q at q (this set corresponds to {ϕ = 0}). For any integers k ≤ ℓ, we write R k,ℓ = ℓ m=k T m (R 0 ) and ξ ℓ k for the set of connected components of M \ R −ℓ,−k . Due to the hyperbolic properties of T , it is easy to see that (see for example [18,Lemma A.1]) We recall that T is discontinuous but 1 2 -Hölder continuous on each connected component of M \ R −1,0 .
1.2. Lorentz process. To avoid ambiguity, at collision times, we only consider reflected vectors. The set of configurations is then The Lorentz process is the flow (Y t ) t defined on M such that, for every (q, v) ∈ M, Y t (q, v) = (q t , v t ) is the couple position-speed at time t of a particle that was at position q with speed v at time 0. This flow preserves the measure ν on M, where ν is the product of the Lebesgue measure on Q and of the uniform measure on S 1 . This flow is naturally identified with the suspension flow (Ỹ t ) t over (M, µ, T ) with roof function τ . Indeed, we recall that (Ỹ t ) t is defined byỸ t (x, s) = (x, s + t) on the set The flow (Ỹ t ) t preserves the measureν onM given by dν(x, s) = dµ(x)ds. Now, we define We have 1.3. Billiard system with finite measure. We defineM as the set of (q, v) ∈ M such that q ∈ I i=1 ∂O i . A point ofM is now parametrized by (i, r, ϕ). We consider the transformation T :M →M , corresponding to T modulo Z 2 . More precisely, if T (q, v) = (q ′ , v ′ ), thenT (q, v) = (q", v) with q" ∈ (q ′ + Z 2 ) ∩ ∪ I i=1 ∂O i . This transformationT preserves the probability measurē µ of density cos(ϕ)/(2 i |∂O i |) with respect to drdϕ.
We call toral billiard system the probability dynamical system (M ,μ,T ).
It is easy to see that (M, µ, T ) corresponds to the cylindrical extension of (M ,μ,T ) by Ψ : M → Z 2 given by Ψ = (S 1 ) |M (with S n defined in the introduction). Indeed

More generally we have
(5) Observe that n−1 k=0 Ψ•T k = S n onM . We recall the following local limit theorem with remainder term. We set β := Note that, if we suppose n ≥ 3k, we can replace the conclusion of this result by Remark 6. Observe that, since the billiard system (M ,μ,T ) is time reversible, if A ⊆M is a union of components of ξ k −∞ and B ⊆M is a union of components ξ k −k , if n > 3k then we have Estimates (6) and (7) will be enough most of the time but not every time. We will also use the following refinements of the local limit theorem.
Proposition 7 (Proposition 4 of [16]). Let any real number p > 1. There exist a 0 > 0 and K 1 > 0 such that, for any integers k ≥ 0, n ≥ 1, any measurable set A ⊆M union of elements of ξ k 0 , any measurable set B ⊆M union of elements of ξ +∞ 0 , for any N ∈ Z 2 , we have We generalize this result as follows.
Proposition 8. Let any real number p > 1. There exist C > 0, a 0 > 0 and K 1 > 0 such that, for any integers k ≥ 0, n ≥ 1 such that n ≥ 4k, any measurable set A ⊆M union of elements of ξ k −k , any measurable set B ⊆M union of elements of ξ +∞ −k , for any N ∈ Z 2 , we have Proof. Observe thatT −k A is a union of elements of ξ 2k 0 and thatT −k B is a union of elements of ξ +∞ 0 . We havē and using the fact that n − 2k ≥ n/2, we obtain the result.

Proof of Theorem 1
Observe that the trajectory of the particle (starting from M ) up to the n-th reflection is Proof of Theorem 1. It follows directly from (8) and from Proposition 10. Indeed Before going into the proof of Proposition 10, let us see the common points betweenV n and V n and let us also explain why the study of V n requires more subtle estimates than the study ofV n . Recall thatV n = n k,j=1 This expression may appear similar to (8), but E 0,k is more complicate thanÊ 0,k . Indeed, inM , we have The union on N is not a problem (it is a finite union since the horizon is finite), the main problem is that the union on x is not finite. Indeed the set V (x) depends on x (and not only on the obstacle containing x).
We recall that, due to Young's construction, if f is constant on each element of ξ N 0 , then there exists a measurablef defined onM such that Let P be the transfer operator on L q of f → f •T seen as an operator on L p . Young proved the quasicompacity of this operator P on V. As in [16], we consider here an adaptation of the construction of Young's towers such that 1 is the only dominating eigenvalue of P on V and has multiplicity one. Hence, there exist K 0 > 0 and a > 0 such that Thanks to this property, Young established an exponential rate of decorrelation. Let us consider Ψ :M → Z 2 the cell-shift function. Recall that, onM , S n = n−1 k=0 Ψ •T k . Since Ψ is constant on each element of ξ 1 0 , there existsΨ :M → Z 2 such thatΨ •π = Ψ • π and the coordinates of Ψ are in V (β 0 ,0) with norm less than 3β −1 0 Ψ ∞ . For any u ∈ R 2 , we define P u (f ) = P (e i u,Ψ f ). Observe that withŜ n := n−1 k=0Ψ •T k . In [20], Szász and Varjú applied the classical Nagaev-Guivarc'h method [12,13,11] to this context. This method plays a crucial role in the proof of Proposition 12 and gives in particular the following inequalities (see [20] and Lemma 12 of [16]) The following result generalizes Proposition 3 of [16].

Estimate of the variance of V n
Recall that Σ 2 is invertible. In particular, there existsã 0 such that (Σ 2 ) −1 x, x ≥ 2ã 0 |x| 2 for every x ∈ R 2 . Comparing we obtain the following useful formula Proof of Proposition 3. As in [1], the proof of Proposition 3 is based on the following formula Var(V n ) = 4
-Using the fact that |x| ≤ 2 min(r, ℓ, s) S 1 ∞ , we get that the contribution to A 2 of the term coming from the composition of the three error terms (e 1 , e 2 , e 3 ) is bounded by if we take p > 1 small enough.
-Now, the contribution to A 2 of the composition of two dominating terms and of one error term of (31), (33) and (33)  On the one hand, we have x e −a 2 0 |x| 2 r+ℓ rℓ ≤ min(r, ℓ) (using the fact that rℓ/(r+ℓ) ≤ min(r, ℓ)). On the other hand, this sum is in O(s 2 ). Therefore the quantity we are looking at is less than if p > 1 is small enough. -If r ≤ 4k or ℓ ≤ 4k or s ≤ 4k, then x is a sum over |x| ≤ 4k S 1 ∞ and one of the following sets is ξ 5k −5k -measurable: We then apply (6) accordingly and take in account the fact that the sum on r or k or ℓ must be taken on {1, ..., 4k}. This leads to a term in o(n 2 ).
-Finally, the estimate of (30) follows the same lines as the estimate of (29). We obtain an analogous estimation multiplied by δ k . This ensures that the contribution of (30) to A 2 is in o(n 2 ). • Control of A 3 . We have This part is the most delicate. Indeed the terms k 1 +r+ℓ+s≤nμ (E 0,r+ℓ+s )μ(E 0,ℓ ) and k 1 +r+ℓ+s≤nμ (E 0,r+ℓ+s ∩ E r,r+ℓ ) are in n 2 log n. But we will prove that their difference is in n 2 . More precisely, we show that First, according to Proposition 10, we have Indeed, setting q = ℓ + r and t = ℓ + r + s, we have Now, let us estimate k 1 +r+ℓ+s≤nμ (E 0,r+ℓ+s ∩ E r,r+ℓ ) in terms ofμ(Ẽ ℓ,C ). For any C, C ′ ∈ P m , we approximate once again C ′ ∩ E 0,r+ℓ+s ∩T −r C ∩ E r,r+ℓ bỹ the measure of which is x H r,ℓ,s,C,C ′ ,x (with x being taken on the set of x ∈ Z 2 such that |x| ≤ min(r, s + 2) S 1 ∞ ) and with H r,ℓ,s,C,C ′ ,x := Now, applying (6) and (7) (when min(r, s) ≥ 3k), we obtain that this quantity is equal toμ the error terms being estimated by .
We obtain that the contribution to A 3 of the dominating terms of (36), (39) and (40) is (where * stands for the sum over k 1 ≥ 1, ℓ ≥ 1 and min(r, s) ≥ 3k) For the third line, we used the fact that Cμ (Ẽ ℓ,C ) = c 2ℓ + O(ℓ −1−η ). For the last line, we used the Lebesgue dominated convergence theorem and the following equalities obtained by a change of variable (r = u + w, s = u + v + w) and by integrating in t, u, r and finally in s: (1 − s) du dr ds rs Now, it remains to show that the contribution to A 3 of all the other terms is in o(n 2 ).
-According to (22), (39) and (40), the contribution of the term coming from the composition of the two error terms e ′ 1 and e ′ 2 is in which is not enough to conclude. Hence, we use the estimate of e ′ 2 given by Remark 9 for x ≥ 3k. On the one hand, the last term in the RHS of the formula given in Remark 9 brings (41) with s -The contribution of the term coming from the composition of the error term e ′ 1 of (39) and of the dominating term of (40) is in if p > 1 is small enough. -For the control of the sum over (k 1 , r, s, ℓ) such that min(r, s) < 3k, we proceed as we did for A 2 . -It remains to estimate The dominating terms obtained by (6) are estimated as the dominating terms of (39) and(40). They bring a contribution to A 3 in The fact that the other terms are in o(n 2 ) follows as for the study of (37).

Proof of Theorem 2
Corollary 15. Let P be a probability measure onM with density h with respect toμ. Assume that h is in L 2 (μ). Then E P [V n ] = cn log n + O(n).
Proof of Corollary 15. We have according to Theorem 3. We conclude thanks to Theorem 1.
Due to the Bienaymé-Chebychev inequality and to the Borel Cantelli lemma, this implies theμalmost sure convergence of (V . Since lim n→+∞ (n + 1) 2 ) = 1, we conclude theμ-almost sure convergence of (V n /(n log n)) n to c.
For any t > 0, we write n t for the number of reflection times before time t. Recall that (t/n t ) t convergesμ-almost surely to Eμ[τ ] as t goes to infinity. Hence we have,μ-almost surely, , as t → +∞.